Complementary metal-oxide-semiconductor (CMOS) circuits have been employed to mimic synaptic Ca2+ dynamics, but three-terminal devices bear limited resemblance to bio-counterparts at the mechanism level and require significant numbers and complex circuits to simulate synaptic behavior. A substantial reduction in footprint, complexity and energy consumption can be achieved by building a two-terminal circuit element, such as a memristor directly incorporating Ca2+-like dynamics. Various types of memristors based on ionic drift (drift-type memristor) have recently been utilized for this purpose in neuromorphic architectures. Although qualitative synaptic functionality has been demonstrated, the fast switching and non-volatility of drift memristors optimized for memory applications do not faithfully replicate the nature of plasticity. Similar issues also exist in metal-oxide-semiconductor-based (MOS-based) memristor emulators, although they are capable of simulating a variety of synaptic functions including spike-timing-dependent plasticity (STDP). Recently, researchers and previous implementations adopted second-order drift memristors to approximate the Ca2+ dynamics of chemical synapses in biological structures by utilizing thermal dissipation or mobility decay, which successfully demonstrated STDP with non-overlapping spikes and other synaptic functions. This previous approach features repeatability and simplicity, but the significant differences of the physical processes from actual synapses limit the fidelity and variety of desired synaptic functions. A device with similar physical behavior to the biological Ca2+ dynamics would enable improved emulation of synaptic function and broad applications to neuromorphic computing.